mean curvature (plane curve)

Let $\Gamma$ be a piecewise $C^{1}$ planar curve.

The total curvature, $\kappa_{total}$ , of $\Gamma$ is defined to be $\int_{\Gamma}|\kappa(s)|ds$ where $\Gamma$ is parameterized by arclength $s$ and $\kappa(s)$ is the curvature (http://planetmath.org/CurvatureOfACurve) of $\Gamma$.

The mean curvature of $\Gamma$ is defined to be the ratio of the total curvature to the length of $\Gamma$ :

 $M(\Gamma)=\frac{\kappa_{total}(\Gamma)}{L(\Gamma)}$
Title mean curvature (plane curve) MeanCurvatureplaneCurve 2013-03-22 15:31:16 2013-03-22 15:31:16 Mathprof (13753) Mathprof (13753) 11 Mathprof (13753) Definition msc 53A04 MeanCurvatureAtSurfacePoint total curvature mean curvature